if the gun is new you should know that there is a break in period where accuracy suffers POI shifts.

Mine likes to have the oil in the barrel cleaned out to nearly nothing. Also I went to a gun show that had a guy really into air guns. He sold me a tin of pellets that are awe inspiring. I am still working through that can. I have killed enough grackles to fill the bed of my pickup. They don't look near as impressive as the name brand ones I was using, but they shoot rings around them. Call one of the airgun rebuilders or custom builders and see what they recommend for your particular gun.

Spring powered air rifles can be quite picky about pellets, but even more so about how you hold them. Experiment with holding your forearm at different points and with minimum pressure at the shoulder and with your trigger hand. Just let it lay there supported mostly by your foreward hand so it can recoil freely. They call it the "Artillery hold" because the rifle is allowed to slide like a big artillery piece. There's no recoil to speak of so just let it move. Try for consistency in your hold, then try different holds as well as different pellets. Some of these can be quite accurate with the proper pellet and treatment.

Each pellet rifle seems to be a law unto itself as to pellet choice, but a general rule of thumb is that pointed pellets give less accuracy than round nose pellets do and flat nose (wadcutter) pellets don't do good in high powered rifles. The Crosman round nose can shoot OK in some rifles. JSB gets high marks from a lot of shooters. I've never had much luck with H&N, but others love them. In the meantime you're spending time getting to know the rifle and what it likes (or doesn't) and it's going to get smoother and smoother. Depending on which rifle you've got, a GRT-III or another of Charlie Da Tuna's triggers can make a world of difference. The one I put in my Gamo was well worth the $30 or so it cost.

They are different from powder burners, that's for sure. But once you find the right combination you'll have a dandy plinking/pesting implement.